Prime avoidance theorem: Let $A$ be a ring (commutative with unity) and $p_1,...,p_n\subset A$ prime ideals. Let $a\subset A$ be an ideal such that $a\subset (p_1\cup p_2\cup\cdot\cdot\cdot\cup p_n)$, then $a\subset p_k$ for some $1\leq k\leq n$.
Now, I have no problem in proving this theorem. I want to illustrate with an example, the importance of the prime condition in the theorem. That is, I want to show that there exist ideals $a_1,...,a_n\subset A$ such that $a\subset (a_1\cup a_2\cup\cdot\cdot\cdot\cup a_n)$, but $a\not \subset a_k$ for all $1\leq k\leq n$. Some of the things that I noticed is that, we cannot find the example in a principal ideal domain nor can we get the example if we take $n=2$. I wasn't able to make much progress beyond this.
Thank you in advance!